And that's the whole point! Your calculation does not say that the integral is a constant, it says the the difference between the two integrals is a constant. Integrals of the same function can differ by a constant.

What you wrote"

is wrong because you have not included the constant of integration you have in your last formula. What you should have is

where and are the constants of integration. That reduces to the statement that

and there is nothing peculiar about that. The two methods just use different constants of integration.

well this was instructive. all that comes to mind is 'oops' .
But it's rather sneaky in that those constants would only appear after the evaluation of , which in your explanation is implicit. that is you're anticipating their arrival because you know one can't subtract indefinite intregals from each other without first accounting for the constants. Thanks!

This example gives me a new appreciation for the constants. i ve always framed them as numbers that'd ensure initial conditions would be met, so as to pin down the exact antiderivative from the continuum of antiderivatives to choose from when evaluating indefinite integrals. In this case, they do more than just that: they show that two methods are compatible but that one of them is actually the only way one will get a useful answer. this last point is nothing new, especially when using integration by parts.
thanks again.